Answer to Problem of the Month for September 2023

Last month, the Girard family moved next to the de Marco's. One fine afternoon, Mrs. Girard called on Mrs. de Marco to get acquainted with the de Marco family. In the course of the conversation, when Mrs. Girard queried Mrs. de Marco about the age of her daughter Jackie, Mrs. de Marco responded cryptically, "When she was born, I was precisely five times the age of her brother Tony at that time. Six years from now, Jackie will be one third of the age which I will attain at that time."

At this point, Mrs. Girard requested some additional information when Mrs. de Marco replied, "The sum of the squares of the digits of my husband's present age in years, is two more than my current age. Presently, my age is nine years more than that of Tony's father (my husband) when Tony was born."

Determine the current ages of Mr. de Marco, Mrs. de Marco and their two children from the above statements.

Solution:

Mr. de Marco is 45 years old.
Mrs. de Marco is 39 years old.
Jackie is 9 years old.
Tony is 15 years old.

EXPLANATION:

Let the respective ages (in years) of Mrs. de Marco, Mrs. de Marco, Jackie and Tony be M, F , G and B respectively.
We denote the sum of squares of the digits of a given number by s.s.d.

Then, by conditions of the problem:

(1) M - G = 5 (B - G);

(2) M +9 = F - B;

(3) M + 6 = 3 (G+6)

(4) s.s.d. (F) = M +2

M - 3G = 12 ( from (3) ) and M + 4G = 5B (from (1))

This gives, 5B - 7G = 12-----------(5)

Since (B,G) = (8,4) is an integer solution for (5), we can take B = 7k + 8, where k must be a non negative integer, giving G = 5k + 4; so that M = 3G +12 = 15k + 24.

Accordingly, from (2); F = M+B- 9; giving, F = 22k + 23, so that:

(M, F) = (15k +24, 22k +23); where k is a non negative integer.

Consequently, (M, F) = (24,19); (39, 41); (54, 63); (69, 85) and (84, 107) are the only available choices keeping in mind the normal life span of a human being. Of the above choices, only (M, F) = (39, 41) satisfies condition (4), since:

4^2 + 5^2 = 41 = 39 +2.;

so that k = (39-24)/15 =1

It can easily be verified that Condition (4) is not satisfied for any other choices of (M, F).

Therefore, G = 5*1 + 4 = 9 and B = 7*1 + 8 = 15. Consequently, the respective ages of Mr. de Marco, Mrs. de Marco, Jackie and Tony are 45 years, 39 years, 9 years and 15 years.

Click here for Carlos de Armas' excellent solution

Correctly solved by:

1. Rob Miles	Northbrook, Illinois, USA
2. Dr. Hari Kishan	D.N. College, Meerut, Uttar Pradesh, India
3. Sudhir Bavdekar	Mumbai, India
4. Kenneth Bennett	Richmond, Virginia, USA
5. Carlos de Armas	Barcelona City, Spain
6. Seth Cohen	Concord, New Hampshire, USA
7. Davit Banana	Istanbul, Turkey
8. Kelly Stubblefield	Mobile, Alabama, USA
9. Ivy Joseph	Pune, Maharashtra, India

September 2023 Problem of the Month

Correctly solved by:

September 2023
Problem of the Month